Air France is operating 4 A319ER for its "Dedicate" Service. They have a configuration of 28J/54Y.
The longest flight they operate is CDG-PNR ( 6032km/3748mi/3257nm) and this is close to their limit when they are full loaded.

The range record for that type of a/c was reached last October 9th 2004, when an A319CJ flew NONSTOP from TLS (Toulouse - France) to LAS (Las Vegas - USA) which means a distance of 9101km ( 4914nm / 5655mi ).

The a/c had 25 persons on board including the Crew and was also carrying 2 tons of freight.

I found 2 sources on the Internet. One says the range is 8300 km, and the other claims it to be 6500 km.

Source 1: http://airbus-a320.biography.ms
The A319LR, compared to the A319CJ has four auxiliary fuel tanks instead of six. Typical range is 4,500 nautical mile(8300km), making it longest range airliner in the A320 family.

Source 2: Air France

Addition of two extra fuel tanks in the rear baggage hold gives it a range of 3,600 nautical miles (6,500km). Air France refers to this A319 as A319-100ER.

Download the PDF file. It has some interesting information about the A319ER.